There are two variants of the 6.5X55 ignoring 19th and early 20th century loadings as used by the the two countries (Norway and Sweden) that jointly developed the cartridge and used in their military weapons.
As a European designed cartridge, the CIP which is the European equivalent of SAAMI
https://en.wikipedia.org/wiki/Commi...nte_pour_l'Epreuve_des_Armes_à_Feu_Portatives
in effect 'owns this design. It calls the cartridge the 6.5X55mm SE and specifies an MAP of 3,800 bar (55,114 psi) but with the rider that this pressure ceiling is for modern firearms, not 19th century designs such as the M1894/96/1938 Swedish Mauser and the various 1880s Norwegian Krag-Jorgensen, not to mention the US manufactured Remington rolling block single shot models.
The US SAAMI organization also lists the cartridge (as 6.5X55 Swedish) and produces specs for it as it was a chambering in an American firearms design, but works on the lower pressures of 46,000 CUP (Copper Units of Pressure) using the older and now discontinued copper pellet crusher pressure gun system. This is roughly equivalent to 51,000 psi using more modern methods.
(From my experience of US deerhunting 140gn PSP Load put over a chronograph, I'd say the larger US ammunition companies are loading to nowhere near 50,000 psi, obviously afraid of it being shot in 100 year old surplus rifles, some possibly in junker condition.)
So one sometimes sees two sets of data, 'SE' and 'Swedish' (or 'Swedish Mauser'), but more often not in which case it is difficult to know which of the two standards is being used unless the text specifies this explicitly.
There is also a third European specification / name - the 6.5X55mm SKAN. (SKAN for Scandinavian) which has caused all sorts of confusion and heartache on this and other forums. From a handloading point of view, this is the 'SE'. The change arose from the Scandinavian countries which have their own keenly fought inter-country match disciplines deciding to select and adopt a single rifle model to be used by all competitors for these matches which also see the host country supply FMJBT arsenal type ammunition to all entrants. It was found that due to slightly different interpretations of the original 1894 case and chamber dimensions' tolerances by Norway and Sweden, it was possible for slight cartridge to chamber mismatches to occur which could allegedly favour the 'home team' in some instances. Therefore as part of the specification process for the new rifle (the Sig-Sauer STR 200 adopted around 1990) a common set of more tightly toleranced dimensions was adopted for both chamber and case.
From a handloading point of view, none of these affects the mechanical / physical aspects. That is die, brass, and bullet specifications are common to all three - it's just (JUST ??) the amount of powder it's safe to use.
I can't help you with Re23 - it's not imported into the UK by Alliant ATK (which also applies to every other recent Swedish manufactured introduction, a shame given the high praise we see on the Accurate Shooter forum for AR-Comp, Re16 and 23.)